8.2 Euclidean Geometry

EUCLID OF ALEXANDRIA

Although he was not responsible for all of the content in The Elements,
Euclid broke new ground in his organization of the foundational
mathematical knowledge of the day.

Euclid is perhaps the most influential figure in the history of mathematics, so it
is somewhat surprising that almost nothing is known about his life. The little
that is known is mainly about his work as a teacher in Alexandria during the
reign of Ptolemy I, which dates to around 300 BC. This was some while after the
creation of Euclid's most famous work, The Elements.

Euclid himself was known primarily for his skills as a teacher rather than for
his theorizing and contributions to research. Indeed, much of the content of the
thirteen volumes that make up The Elements is not original, nor is it a complete
overview of the mathematics of Euclid's time. Rather, this text was intended
to serve as an introduction to the mathematical concepts of the day. Its great
triumph was in presenting concepts in logical order, beginning with the most
basic of assumptions and using them to build a series of propositions and
conclusions of increasing complexity.

AXIOMATIC SYSTEMS

Axioms are agreed-upon first principles, which are then used to generate
other statements, known as "theorems," using logical principles.

Systems can be internally consistent or not, depending on whether or not
their axioms admit contradictions.

The system that Euclid used in The Elements—beginning with the most basic
assumptions and making only logically allowed steps in order to come up with
propositions or theorems—is what is known today as an axiomatic system. Here
is a very simple example of such a system:

Given the things: squirrels, trees, and climbing,

There are exactly three squirrels.

Every squirrel climbs at least two trees.

No tree is climbed by more than two squirrels.

A logical theorem could be the statement: there must be more than two trees.

A simple picture would prove this theorem:

So, a theorem is something that can be shown to be true, given a set of basic
assumptions and a series of logical steps with no contradictions introduced.

Now, consider the following axiomatic system:

Given the things: cat, dog

A cat is not a dog.

A cat is a dog.

It is clear that both statements 1 and 2 cannot be true simultaneously. However,
these are the basic axioms of our system, and axioms have to be assumed
to be true—so, this system is clearly worthless, because it contains a logical
contradiction from the start. In other words, it is not self-consistent. In this
example, the contradiction presents itself directly in the axioms, but most
contradictory systems are not so easy to identify.

FOUNDATIONS OF GEOMETRY

Euclid used five common notions and five postulates in The Elements.

The fifth postulate, also known as the "parallel postulate," is somehow not
like the others.

When Euclid laid the foundation for The Elements, he had to be careful to start
with statements that would be both self-consistent and basic enough to be
assumed true. He divided his initial assumptions into five postulates1 and five
common notions. (Note: A postulate is not quite the same as an axiom. Axioms are general
statements that can apply to different contexts, whereas postulates are
applicable only in one context, geometry in this case.) They are as follows:

Common Notions:

Things that are equal to the same thing are also equal to one another.

If equals be added to equals, the wholes are equal.

If equals be subtracted from equals, the remainders are equal.

Things that coincide with one another are equal to one another.

The whole is greater than the part.

Postulates:

Any two points can be joined by a straight line.

Any straight line segment can be extended indefinitely in a straight line.

Given any straight line segment, a circle can be drawn having the segment
as radius and one endpoint as center.

All right angles are congruent.

If two lines intersect a third in such a way that the sum of the inner angles
on one side is less than two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough.

That fifth postulate is a mouthful; fortunately, it can be rephrased. In the
fifth century, the philosopher Proclus re-stated Euclid's fifth postulate in the
following form, which has become known as the parallel postulate:
Exactly one line parallel to a given line can be drawn through any point not on
the given line.

This postulate is somehow not like the other four. The first four seem to be
simple and self-evident in that it seems things could be no other way, but the
fifth is more complicated. Euclid, himself, likely noticed this discrepancy, as
he did not use the parallel postulate until the 29th proposition (theorem) of The
Elements.

Euclid's system has been incredibly long-lasting, and it is still standard fare
in high school geometry classes to this day. It represents an achievement in
organization and logical thought that remains as relevant today as it was 2000
years ago. That bothersome fifth postulate, however, showed a small crack
in the foundation of the system. This crack was ignored for centuries until
mathematicians of the 1800s, with further exploration, found it to be a doorway
into a world of broader understanding.